The semantics of many, much, few, and little∗

Jessica Rett

September 2017

Abstract

The words many, much, few and little (and their cross-linguisticcounterparts) are quite unusual semantically. They have traditionallybeen characterized as quantifiers (like every) or adjectives (like tall);however, these analyses can only account for instances of these terms inwhich they encode information about an individual or a set of individ-uals, as they do when they occur pronominally (in e.g. much traffic).Recent degree-semantic analyses (Rett, 2007, 2008b, 2014; Solt, 2009,2015) instead characterize the meaning of these words in terms of in-tervals, or sets of degrees; this accounts for their canonical uses as wellas uses in which they don’t appear to be ranging over individuals (asin their differential use, e.g. much taller).

1 Introduction

The words many, much, few and little have received several very differentsemantic treatments in the literature. This arguably due to the fact thatthey are so versatile; they appear to be quantifiers, adjectives, or modifiers,depending on which of their uses are taken to be canonical. The goal of thispaper is to provide a comprehensive view of the meaning and distributionof these words, and to review and evaluate semantic treatments of them.

2 An empirical overview

The semantic status of many, much, few and little is so indeterminate thatthere is no standard term for them and their cross-linguistic counterparts.

∗Thanks to Brian Buccola, Robert Henderson, Alex Hughes, Ed Keenan, RogerSchwarzschild, and the students in my Spring 2016 seminar at UCLA for discussion ofthese issues. The paper has benefitted immensely from the helpful and careful advicefrom two anonymous reviewers.

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I will settle on the relatively theory-neutral term “quantity words”.

2.1 The meaning(s) of quantity words

Quantity words are associated with a number of different semantic prop-erties. This section reviews the properties that have received the mostattention in the theoretical literature: the meaning of negative quantitywords; the cardinal/proportional ambiguity; and the evaluative, ‘above-the-standard’ meaning of some constructions that contain quantity words.

2.1.1 Differences between quantity words

It’s generally assumed that the difference between many/few on the onehand and much/little on the other corresponds to the difference betweencount and mass nouns. While it is unclear what that distinction is (seeChierchia, 2010, for an overview), a common working hypothesis is thatmany and few are associated with discrete, countable entities whose quan-tities can be measured with whole numbers, while much and little are as-sociated with non-discrete entities whose quantities must be measured withrational numbers. A consequence, further discussed in §2.2, is that the dis-tribution of much and little is broader than that of many and few : theformer, but not the latter, can measure the quantity of events and abstractintervals as well as individuals.1

Many languages don’t lexicalize a count/mass distinction in their quan-tity words (e.g. Romanian, Rett 2007), but they do generally lexically dis-tinguish between positive (e.g. many, much) and negative (e.g. few, little)quantity words. The polarity distinction has, consequently, received a fairamount of attention in the theoretical literature, which generally assumesthat e.g. many and few are antonyms, just like big and little. Negativeantonyms are widely viewed as marked relative to their positive counter-parts; i.e., a negative antonym like little is the negated version of its posi-tive counterpart much, but not vice-versa (Lehrer, 1985; Rett, 2015). Heim(2007) makes this point explicitly for little, arguing that it involves a seman-tic negation that can scope above or below a modal.

1However, the distribution of much in English is subject to additional restrictionsrelative to the other quantity words: it is only acceptable outside of negation in some(e.g. Much attention was paid to this document) but not all contexts (e.g. *John ate muchrice). Israel (1996) identifies much as the paradigm case of a so-called ‘attenuating NPI’.

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2.1.2 Cardinal and proportional readings

Many constructions containing prenominal quantity words can receive a car-dinal or at least one proportional reading (Milsark, 1977; Westerstahl, 1985;Partee, 1989; Buring, 1996; Herburger, 1997). This is demonstrated in (1),where S denotes the set of Scandinavians, W the set of Nobel Prize winners,and where d ranges over some contextually salient quantity threshold.

(1) Many Scandinavians have won the Nobel Prize in literature.

a. |S ∩W | ≥ d cardinalb. |S∩W|S| | ≥ d proportional

c. |S∩W|W | | ≥ d reverse proportional2

In (1-a), the number of Scandinavians who are Nobel Prize winners isunderstood to be a large number. In (1-b), the number of Scandinavianswho are winners is understood to be large relative to the number of Scan-dinavians. Finally, in (1-c), the number of Scandinavians who are NobelPrize winners is understood to be large relative to the number of NobelPrize winners generally. As Partee observes, the proportional readings canbe paraphrased with partitive sentences: Many of the Scandinavians areNobel Prize winners (for the proportional reading) and Many of the NobelPrize Winners are Scandinavians (for the reverse proportional reading).

Theoretical treatments of the semantics of quantity words, discussed fur-ther below, vary in their treatment of these readings. Earlier analyses (e.g.Partee, 1989) treated them as an ambiguity; Herburger (1997) argues thatsyntax and focus condition the availability of the three interpretations. Re-cent semantic analyses have tried to derive the cardinal and proportionalinterpretations from a single underspecified, context-sensitive lexical entry,both in quantificational accounts (Westerstahl, 1985; Lappin, 2000; Cohen,2001; Romero, 2015) and non-quantificational accounts (Rett, 2008b; Solt,2009). The availability of the ‘reverse proportional’ interpretation meansthat, if quantity words denote quantifiers, they violate conservativity (West-erstahl, 1985); this has lead those working in a quantificational frameworkto conclude that quantity words require an intensional analysis (Fernandoand Kamp, 1996).

2As Herburger (1997:67) notes, this reading is available when Scandinavians is focused.

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2.1.3 Quantity words and evaluativity

Quantity words, like adjectives, participate in evaluative (or ‘norm-related’)constructions. A construction is evaluative iff it requires of some degree thatit exceed a contextually-valued standard of evaluation (see Rett, 2015, foran overview of this phenomenon).

Positive constructions formed with quantity words and relative adjectives– constructions in which these words occur unmodified and unbound – arealways evaluative, as demonstrated in (2). (2-a) is true only if the heightof the students in question exceeds the contextual standard for tallness (orshortness). Similarly, (2-b) is true only if the quantity of students in questionexceeds the contextual standard for many (or few).

(2) a. Tall/Short students received new desks.b. Many/Few students received new desks.

However, not all of the constructions that quantity words and relativeadjectives occur in are evaluative. This is demonstrated below for the equa-tive construction.

(3) a. Mary is as tall as John... which is to say, she’s short.b. Mary is as short as John... #which is to say, she’s tall.

(4) a. Mary teaches as many students as John... which is to say, fewstudents.

b. Mary teaches as few students as John... #which is to say, manystudents.

Neither of the initial sentences in (3-a) and (4-a) requires that the degree inquestion exceed a contextually-valued standard, as evidenced by the accept-ability of the antonymic continuations. In contrast, the negative-antonymsentences in (3-b) and (4-b) are evaluative, as evidenced by the unaccept-ability of the antonymic continuations. But not all constructions contain-ing negative-antonym quantity words and relative adjectives are evaluative;comparatives formed from negative quantity words and relative adjectivesare not evaluative, as demonstrated in (5).

(5) a. Mary is shorter than John... but of course, she is still tall.b. Mary teaches fewer students than John... but of course, she still

teaches many students.

These data have lead semanticists to draw the same conclusion aboutquantity words as they have about relative adjectives (Cresswell, 1976; Bier-

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wisch, 1989; Rett, 2008a): an empirically comprehensive theory of evaluativ-ity requires the assumption that evaluativity is not lexically encoded in themeaning of quantity words themselves, but that it arises for external rea-sons, only in certain configurations. This conclusion runs contrary to early,quantifier-based treatments of quantity words that encode evaluativity inthe lexical entry of quantity words.

2.2 The distribution of quantity words

Quantity words have a very wide distribution across constructions and phrasalcategories. The canonical use of quantity words is their prenominal or at-tributive use, as in (6).

(6) a. Many/few guests left.b. Much/little rice was consumed.

It is this use that makes it tempting to classify them as quantifiers, giventhe parallel data in (7-a), although (7-b) shows that adjectives can occur inthis position as well.

(7) a. All/some guests left.b. Tall/short guests left.

Quantity words in this prenominal position can combine with demon-stratives like that and wh-phrases like how and what (although the latteronly in exclamatives, Rett 2009). Many, few and little can additionally bemodified by the determiner the, and few and little can additionally be mod-ified by the indefinite determiner a. Quantity words share these propertiesability with adjectives, but not with canonical quantifiers, as demonstratedin (8) and (9).

(8) a. *John didn’t meet that all/some guests.b. John is not that tall/short.c. John didn’t each that much/many cookies.

(9) a. *The all/some guests left.b. The tall/short guests left.c. The many/few guests left.

Also like adjectives, but unlike quantifiers, quantity words can occur inpredicative position, as shown in (10), although the English mass quantitywords much and little are generally reported to be less acceptable in thisposition.

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(10) a. *John’s regrets are all/some.b. John’s children are tall/short.c. John’s regrets are many/few.

There are other canonically adjectival positions in which quantity wordscan occur: those described by Bresnan (1973) as involving degree quantifierslike the equative as, the excessive too, and so.

(11) a. John is as tall as Sue.b. John has as many regrets as Sue.

(12) a. John is too irresponsible to be an assistant.b. John has too much experience to be an assistant.

(13) a. John is so clumsy that he can’t play tennis.b. John has so few suitcases that he had to borrow mine.

Two other constructions that fit in this paradigm are the comparativemorpheme -er and superlative morpheme -est (Heim, 2000; Schwarzschild,2008). While the forms of these constructions involving few are transpar-ently constructed from quantity words, the other forms (more, most andless, least) very clearly involve suppletion (Bresnan, 1973; Bobaljik, 2012).It should be clear that standard quantifiers like all and some cannot occurin these positions.

From the discussion so far, quantity words pattern with quantifiers inonly one configuration: prenominally. But they pattern with adjectives inseveral (prenominally; in predicate position; with overt determiners; andwith degree quantifiers). Quantity words are, however, acceptable in threeadditional configurations which admit neither quantifiers nor adjectives: themodification of verb phrases (VPs); preposition phrases (PPs); and compar-atives, demonstrated in (14)–(16) below.

(14) a. John doesn’t go to the movies much.b. John sleeps little.

(15) a. The car drove much/little over/under the speed limit.b. The picture isn’t much above the mirror.

(16) a. John is much/little taller than Sue.b. John drove much/little faster than Sue.c. The desk is much/little farther away than the couch.

These distributional patterns are summarized in Table 1.In what follows, I will be making a single distinction within the contexts

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quantifiers adjectives quantitywords

(e.g. all) (e.g. tall) (e.g. many) examples

ind

ivid

ual

prenominally X X X (6)-(7)with a determiner × X X (9)

predicative position × X X (10)degree quantifier × X X (11)–(13)

non

-in

d

VP modifier × × X (14)PP modifier × × X (15)

comparative modifier × × X (16)

Table 1: The distribution of quantity words in English

in which quantity words can occur. ‘Individual uses’ of quantity words arethose in which they appear to be ranging over individuals (or sets of indi-viduals). This category includes the prenominal and predicative contexts,as well as prenominal uses of quantity words in degree quantifier construc-tions. In these cases (e.g. (the) many guests, The guests were many, as manyfriends as) we have an intuition that the quantity word encodes informationabout the measure of an individual or a plurality of individuals.

In contrast are the ‘non-individual uses’ of quantity words, in which wehave an intuition that the quantity word doesn’t encode information aboutan individual. When they function as VP modifiers, quantity words en-code information about the measure of an event (Doetjes, 2007; Nakanishi,2007), e.g. its duration or frequency. When they function as PP modifiers,quantity words encode information about the size of a spatial interval, or avector (Zwarts, 1993). And when they modify comparatives (their ‘differen-tial’ use), quantity words encode information about the size of a differentialinterval along some dimension of measurement (Schwarzschild, 2006a), e.g.height.3

To summarize: quantity words are available in a number of different con-figurations. Their prenominal use, often treated as canonical, has lead someto analyze quantity words as quantifiers, and others to analyze them as ad-jectives, but they pattern like neither in their overall distribution. The rest

3While I categorize degree quantifier examples like too much rice as individual uses, it’simportant to note that quantity words can also occur with degree quantifiers to modifyevents, e.g. go to the movies too much. These instances of degree quantifier constructionswould best be characterized as non-individual uses, along the lines of (14).

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of this article outlines semantic treatments of quantity words as quantifiers(§3) and as adjectives (§4), and argues that these distributional differencesmean that neither sort of theory can accurately characterize the behavior ofquantity words. The details are interesting, but the overall point should beclear, given Table 1: analyzing quantity words as quantifiers predicts thatthey have the same distribution as quantifiers do, which they don’t. Andanalyzing quantity words as adjectives predicts that they have the same dis-tribution as adjectives do, which they don’t. In particular, both approachesincorrectly predict that quantity words only have individual uses.§5 presents recent analyses of quantity words in which they range over

intervals, or sets of degrees. These accounts work well for the diverse datareviewed here because the framework of degree semantics allows for a varietyof different semantic objects (individuals, events, etc.) to be associated withsets of degrees corresponding to their measure in a given context.

2.3 Crosslinguistic differences

Although most of the semantic literature on quantity words focuses on En-glish, recent research has revealed some additional ways in which quantitywords can vary across languages.

Quantity words vary in the extent to which they contribute to quantitywh-words (the equivalent of English how many): in Balkan languages, quan-tity words can optionally occur with quantity wh-words, and effect subtlesemantic differences when present (Rett, 2007, 2008b).

(17) a. Cıtehow.many-f.pl

femeiwomen

cunoa ste?know-3.sg

b. Cıthow.many

deof

multemany-f.pl

femeiwomen

cunoa ste?know-3.sg

‘How many women does he know? Romanian

Some Slavic languages additionally seem to include two sets of quantitywords, even setting aside distinctions of polarity and count/mass. Krasikova(2011) argues that the Russian adverb mnogo can only receive a cardinalinterpretation, while the adjectival mnogie can only receive a proportionalinterpretation. Stateva and Stepanov (2016) present an in-depth look atthe Slovenian quantity words precej and veliko. They argue that, contraryto Russian, the words do not differ in terms of the cardinal/proportionaldistinction, but rather in their distribution (see §2.2): both can combinewith NPs or the superlative, but only precej can combine with adjectives,adverbs, or PPs.

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Finally, there are cross-linguistic differences in the way quantity wordscan range over events. Doetjes (2007, 2008) observes that Standard Frenchbeaucoup can range over events in e.g. J’ai beaucoup dormi (‘I have slepta lot’), while Burnett (2012) observes that beaucoup can additionally rangeover ordered pairs of individuals and events in Quebec French (Nakanishi2007 makes similar observations for Japanese classifier constructions.)

3 Quantificational accounts of quantity words

One interpretation of the term quantifier describes expressions that involvecounting or measuring. This certainly characterizes quantity words. Butthere is a specific use of the term wherein it describes expressions thatdenote relations between sets of individuals, as it does in Generalized Quan-tifier Theory (GQT). In this section, I’ll argue that quantity words cannotaccurately be analyzed as quantifiers in this more specific sense.

3.1 Generalized Quantifier Theory

In GQT (Barwise and Cooper, 1981; Keenan and Stavi, 1986; Greer, 2014),quantifiers like all and some are analyzed as denoting relations between setsof individuals A and B, as (18) shows.4

(18) a. JallK = λAλB[A ⊆ B]b. JsomeK = λAλB[A ∩B 6= ∅]c. JmanyK = λAλB[|A ∩B| ≥ d, d a large number]

(18-a), for instance, predicts that a sentence All ants boogied is true iff the setof ants is a subset of the set of individuals that boogied. An equivalent wayof writing these truth conditions is with an overt logical quantifier rangingover individuals, as in (19) for the quantifier some.

(19) JsomeK = λAλB∃x[A(x) ∧B(x)].

In GQT, and in many logic traditions, quantity words like many areanalyzed in the same fashion, although, as (18-c) shows, they require addi-

4I present all proposed denotations in the lambda calculus, in which each lambda (λ)abstracts over an argument representing an argument of the function denoted by thelexical entry in question. Thus the entry in (18-a) is equivalent to the more traditionalset-theoretic definition in (i).

(i) JallK(A)(B) is true iff A ⊆ B

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tional machinery. In particular, the GQT treatment of many incorporatesthe cardinality operator |.| and a relation to the free variable d, as a way ofencoding evaluativity. (18-c) predicts that a sentence Many ants boogied istrue whenever the cardinality of the intersection of the set of ants and theset of individuals that boogied is greater than or equal to some large num-ber d, which can receive different values depending on the context. Thisappropriately characterizes constructions with prenominal quantity wordsas evaluative and thereby as context-sensitive.

3.2 The Parameterized Determiner approach

Romero (1998) and then Hackl (2000) propose an updated analysis of quan-tity words as quantifiers, based on their observations about the behavior ofquantity words in how many questions (Romero, 1998) and in comparativeslike more (Hackl, 2000). Both (see also Romero, 2015) analyze quantitywords as ‘parameterized determiners’: quantifiers with an additional degreeargument d, as in (20), making them a hybrid between a quantifier and howdegree-semantic analyses treat gradable adjectives (which will be discussedin more detail in §4).

(20) JmanyK = λdλAλB∃x[A(x) ∧B(x) ∧ |x| = d]

Although the parameterized determiner analysis was proposed in con-trast to the GQT analysis, the two have in common that they analyzequantity words as encoding existential quantification (or its set-theoreticequivalent) over individuals. They therefore predict that quantity words aredefined only when they range over individuals. Quite simply, these accountsincorrectly predict that quantity words are undefined in the non-individualuses described above (e.g. the comparative modifier or differential use).

3.3 Discussion

At the very least, then, adopting a quantifier-based analysis of prenominalquantity words requires a separate treatment for their other uses, which sug-gests that it’s an accident prenominal quantity words are homophonous withe.g. differential ones. But there are still more strikes against quantifier-basedapproaches. They cannot account for predicative uses of quantity words, forexample. They incorrectly predict that prenominal quantity words cannotcombine with overt determiners like the, which are also analyzed as rangingover individuals. (If many did in fact bind the individual argument of guestsin The many guests left, we would predict that argument to be unavailable to

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the, resulting in vacuous quantification.) And although quantity words areoptional in how many questions in several Balkan languages (Rett, 2007),parameterized determiner treatments predict that they’re necessary to bindthe individual variable introduced by the nominal. See Rett (2008b) andSolt (2015) for additional worries about these approaches.

These issues raise the question: if quantity words aren’t quantifiers, arethey better analyzed as adjectives?

4 Adjectival accounts of quantity words

Table 1 suggests that quantity words and adjectives mean the same sort ofthing, at least some of the time. This perspective has a modest history ofsupport in linguistic semantics (Bresnan, 1973; Cresswell, 1976; Hoeksema,1983; Grosu and Landman, 1998; Partee, 1989), based on the fact thatquantity words can “...form comparatives and superlatives, they can combinewith degree expressions like too or very.... Many and few can also be used inpredicative position (his sins were many, his virtues were few)” (Hoeksema,1983, 65).

4.1 The semantics of gradable adjectives

If quantity words are adjectives, they are gradable adjectives like tall (asopposed to non-gradable ones like polka-dotted). A gradable adjective ischaracterized by its ability to be modified by intensifiers like very and to oc-cur with degree quantifiers like the comparative, crucially without semanticcoercion to some related scale (as the non-gradable pregnant is often coercedto a temporal scale in e.g. very/more pregnant ; Cruse, 1986). Quantity wordspass both tests (as demonstrated by very few and fewer).

Beginning with Cresswell (1976), degree semantics treats gradable adjec-tives as differing from non-gradable ones in taking a degree argument.5 Inmodern degree semantic approaches, degrees are semantic primitives, rang-ing over points on scales that correspond to different dimensions of mea-surement, like height, happiness, etc. From this perspective, the differencebetween gradable and non-gradable adjectives is one of valency, similar tothe difference between transitive and intransitive verbs, demonstrated in(21). (Furthermore, in degree semantics, intensifiers and degree quantifiersrange over degrees, which explains why they can’t modify non-gradable ad-jectives.)

5See Schwarzschild (2008); Morzycki (2015) for an introduction to degree semantics.

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(21) a. Jpolka-dottedK = λx[x is polka-dotted]b. JtallK = λdλx[x is d-tall]

According to (21-b), the gradable adjective tall relates an individual x to adegree d whenever x is tall to at least degree d. As a result, each individualis potentially associated with a plurality of degrees of tallness, or a degreeinterval. In the case of an individual whose maximum height is 6ft, thisinterval would have an open lower bound of zero and a closed upper boundof 6ft, i.e. {...1ft,...,6ft}, or (0,6ft].

4.2 Adjectival quantity words

In adjectival accounts of quantity words, they, too, take an individual anda degree argument, as in (22).6

(22) JmanyK = λdλx[x is d-many]

As in (21-b), (22) states that many relates a (plural) individual x to adegree d whenever the cardinality of x is at least d. As a result, each pluralindividual is associated with a plurality of quantity degrees, i.e. an interval.In the case of a plural individual with an atomic quantity of 6, this intervalwould be {1,2,3,4,5,6}, or a discrete interval with closed lower bound of 1and a closed upper bound of 6, i.e. {1,...6}, or [1,6].

In these degree-semantic accounts, gradable adjectives can combine withan overt measure phrase, which values their degree argument (e.g. John is6ft tall).7 In the absence of an overt measure phrase or degree quantifier, theextra degree argument must be related to a contextual standard s (becausepositive constructions are evaluative) and existentially bound, as shown in(23) and (24).

(23) JJohn is tallK = ∃d[John is d-tall ∧ d > stall]

(24) JThe guests are manyK = ∃d[the-guests are d-many ∧ d > smany]

There are several assumptions regarding how these bits of meaning enter thederivation: most commonly, via a null operator ‘pos’ (Bartsch and Venne-mann, 1972; Cresswell, 1976; von Stechow, 1984; Kennedy, 1999); but there

6These are the definitions for adjectives in predicative position; those in attributiveposition are assumed to have a related, type-raised definition that includes a set of indi-viduals P as an argument, e.g. Jtall(P )(x)(d)K is true iff x is P and d-tall.

7Quantity words cannot occur with overt numerals or measure phrases (cf. *6 manypizzas), but neither can the vast majority of gradable adjectives (cf. *$20 expensive). Thedifference is generally treated as a lexical rather than semantic one (Schwarzschild, 2005).

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have been other proposals involving other null operators, or conversationalimplicature (Rett, 2008a,b, 2015).

The truth conditions in (23) hold in a situation in which there is somedegree to which John is tall that exceeds on the ‘tall’ scale some contextuallysalient standard of tallness stall. The truth conditions in (24) hold in asituation in which there is some degree corresponding to the quantity ofguests that exceeds on the ‘many’ scale some contextually salient standardof quantity smany.

In these degree-semantic approaches, antonymy is generally treated as areversal in scale ordering (Bartsch and Vennemann, 1972; Morzycki, 2015;Rett, 2015); this accounts for the mutual entailment of sentences like A hasmore than B and B has less than A. Thus the negative-antonym version of(24) would require that there be some quantity of guests that exceeds thecontextually salient ‘few’ standard on the ‘few’ scale. This is equivalent tothe requirement that the quantity of guests be lower than sfew on the ‘many’scale. This is illustrated in Figure 1 below.

Figure 1: Antonymic relations as scale reversals

4.3 Discussion

It should be clear from the discussion above that adjectival accounts, likequantificational accounts, define quantity words in terms of their individualargument. In (22) (and its attributive counterpart), the quantity word manytakes an individual argument, and the semantic contribution of the quantityword is to associate this individual with a degree argument correspondingto its measure. In these adjectival accounts, quantity words are undefined ifthey do not take an individual argument (although see Wellwood, 2015, foran account that re-envisions adjectival meanings in light of this constraint).They therefore cannot extend to the non-individual uses of quantity words,which do not encode information about an individual. In the comparative

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John is much taller than Sue, for instance, the quantity word much intu-itively measures the difference between John’s height and Sue’s height, notJohn or Sue themselves.

As with the quantificational accounts, since adjectival accounts cannotextend to these non-individual uses of quantity words, they are forced toclaim that it is an accident that the same word is used in English to measureindividuals and non-individuals. Worse, they cannot predict that quantitywords have this multiplicity of meaning across languages. What Table 1 sug-gests is that we need an account of quantity words in which their distributionis less semantically restricted.

5 Interval-based treatments of quantity words

If quantity words are neither quantifiers nor adjectives, what are they?Domain-general accounts of quantity words take the perspective that if thenon-individual uses of quantity words can’t be properly treated in individual-based accounts, then the individual uses of quantity words should be treatedin degree or scale-based accounts that are well-suited to treat the non-individual uses. In other words, instead of considering the prenominal orpredicative instances of quantity words to be canonical (like the quantifierand adjectival accounts do, respectively), domain-general approaches treatthe !differential instances as canonical.

5.1 Quantity words as degree modifiers

In a series of proposals (2007; 2008b; 2014), Rett proposed that quantitywords should be recast in terms of intervals, or sets of degrees. Since thereis independent evidence that all sorts of semantic domains – including in-dividuals and events – need to be associated with degrees, the switch fromindividuals to intervals results in a more category-general account of quan-tity words.

5.1.1 Differential quantity words

The underlying intuition is this: in the differential comparative John is muchtaller than Sue, quantity words measure the size of the gap between John’sheight and Sue’s height. This gap is modeled in degree semantics using ascale, i.e. a set of degrees representing points along a single dimension ofmeasurement (Schwarzschild, 2006a), as illustrated in Figure 2 below.

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Figure 2: Scales and gaps in degree semantics

In this account, quantity words are degree modifiers (type 〈〈d, t〉, 〈d, t〉〉),denoting relations between a set of degrees D (i.e. an interval) and its sized, as shown in (25).8

(25) JmuchK = λDλd[d is the size of D]

Intervals are measured by subtracting the lower bound from the upperbound.9 In Figure 2, for example, the measure of the interval (5ft,6ft] –representing the gap between John and Sue’s heights – is 1ft.

A transparent instance of quantity words modifying intervals is their useas modifiers of comparatives, as in John is much taller than Sue is. In thedegree semantics literature, a clausal comparative sentence like (26-a) hasthe logical form in (26-b), with each clausal argument of the comparativedenotes a set of degrees, or an interval (Bresnan, 1973), and the comparativemorpheme -er is a degree quantifier, as in (27).10

(26) a. John is taller than Sue is.b. -er([Opd′ Sue is tall to d′])([Opd John is tall to d])

(27) J-erK = λDλD′λd[D′(d) ∧ ¬D(d)]

In (27), the variable d ranges over differential degrees: those that are inthe the interval associated with the matrix subject but not the intervalassociated with the embedded subject (i.e. the gap between the measures ofthe two subjects).

8The motivation for this characterization, as stated in Rett 2008b, is that quantitywords are like other modifiers in their ability to be reduplicated, as in much, much later.

9In measure theory, the size of an interval with endpoints a and b (where a < b) is b−a.This measure is defined over closed, open, and partially closed intervals alike (Gordon,1994). When intervals are downward monotonic (i.e. when the inclusion of d in the setentails the inclusion of d−1) the measure of an interval is equivalent to its maximal degree,which yields some interesting results for natural language semantics (Rett, 2007, 2008b).

10This is a version of the ‘A-not-A’ analysis, which has its origins in Ross (1969); Seuren(1973); McConnell-Ginet (1973); Kamp (1975); Hoeksema (1983); Seuren (1984) and isreviewed in Schwarzschild (2008).

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In the case of unmodified comparatives, as in (26), the differential degreeargument d is assumed to be existentially bound, resulting in the informaltruth conditions ‘There is some degree d such that John is tall to degree dbut Sue is not.’11

It is this degree argument d that quantity words like much modify ininterval-based approaches: in a sentence like (28), much measures the sizeof the gap between John and Sue’s height.12 In the absence of any additionalmodification (see footnote 8), the degree argument of the quantity word isexistentially bound. And the construction is evaluative as the result ofthe usual mechanism for bestowing evaluativity on unbound adjectives andmodifiers (see §2.1.3). This is depicted in (28-c), where the size d′ of thegap between John’s and Sue’s heights is required to exceed the contextually-valued standard for tallness gaps (st−int).

(28) John is much taller than Sue

a. JmuchK({d : John is d-tall ∧ ¬(Sue is d-tall)})b. = λd′[d′ is the size of {d: John is d-tall ∧ ¬(Sue is d-tall)}]c. = after existential closure, evaluativity∃d′[d′ is the size of {d: John is d-tall ∧ ¬(Sue is d-tall)}∧ d′ > st−int]

The truth conditions in (28-c) characterize the differential comparative asrequiring that the size of the gap between John’s height and Sue’s height(the set of degrees d, e.g. the interval (5ft,6ft], given Figure 2) is consideredbig in the context of utterance.

This interval-based account treats the use of quantity words as compar-ative modifiers as primary. Extending it to cases in which quantity wordsmeasure other sorts of things (i.e. its individual and event uses), requiresexisting mechanisms in degree semantic theories that use null operators ortype-shifters to associate an individual with a degree corresponding to somesalient measure of that individual.

5.1.2 Individual uses of quantity words

In standard degree-semantic accounts, numerals and measure phrases denotedegrees. Without further innovation, this would result in a type mismatch

11This treatment of differential comparatives requires a type-shifted semantics for dif-ferential measure phrases; see Schwarzschild (2005, 2006a) and Morzycki (2008).

12For ease of exposition, in (28) I switch in some cases from the λ-calculus characteristicfunction of a set of degrees (λd′...) to its set-theoretic equivalent ({d′:...).

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in phrases in which numerals modify nouns, like five guests, as nouns tra-ditionally denote sets of individuals. A standard solution, beginning withCresswell (1976), is to allow nouns to be optionally associated with a degree(via a null operator or a type-shifting mechanism, as in (29), Schwarzschild2002, 2005; Nakanishi 2007).

(29) JM-OpK = λPλdλx[P (x) ∧ the salient measure of x is d]

‘M-Op’ relates individuals x in the extension of some predicate P to theirdegree of measurement along some salient dimension (allowing for sentenceslike The board is 3 feet to measure e.g. length or width, depending).13 In thecase of five guests, or any numeral phrase, the only available dimension ofmeasurement is quantity. A sample derivation is in (30); as shown in (30-b),the individual variable is bound via existential closure in the absence of anyovert binding (see Rett, 2014, for details).

(30) Five guests arrived.

a. JM-Op guests arrivedK= λdλx[guests(x) ∧ arrived(x) ∧ the quantity of x is d]

b. Jfive M-Op guests arrivedK = after existential closure∃x[guests(x) ∧ arrived(x) ∧ the quantity of x is 5]

The observed ability of different types of semantic objects to option-ally associate with degrees allows interval-based accounts to generalize fromthe comparative modifier use of quantity words to their other uses. (31)illustrates how Rett’s account of quantity words extends to their canonicalprenominal use. In it, the individual argument undergoes existential clo-sure, creating in (31-a) the characteristic function of the set of quantities ofguests who arrived. As in (31-b), the quantity word measures this set; sincethe quantity word is not bound or modified, its degree variable is assumedto undergo existential closure.14

13Rett (2008b) assumes that M-Op has a predicative formulation, related to the attribu-tive version in (29) via the same type-shifter that associates predicative and attributivegradable adjectives. See Schwarzschild (2006b) and Rett (2014) for arguments that, whilemass quantity words like much can be associated with a variety of different dimensionsof measurement, they can only be associated with dimensions that are monotonic on therelevant part-whole structure of an entity.

14Rett (2007; 2008b; 2014) assumes that, in sentences like (31), the NP subject and VPcombine first via predicate modification before both combine with M-Op. This assumptionensures that the degree measured by many is the number of guests who arrived, not thenumber of guests.

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(31) Many guests arrived.

a. JM-Op [guests arrived]K = after existential closure over xλd∃x[guests(x) ∧ arrived(x) ∧ the quantity of x is d]

b. Jmany [M-Op [guests arrived]]K= after existential closure over d, evaluativity∃d′[d′ is the size of {d: ∃x[guests(x) and arrived(x) ∧ the quan-tity of x is d]} ∧ d′ > sq−int]

The truth conditions in (31) predict that the sentence is true if the size ofquantities corresponding to the plurality of guests who arrived exceeds thecontextually valued standard sq−int for intervals corresponding to quantities(of guests who arrived). Importantly, the size of a downward-monotonic setof degrees corresponds to its maximal degree (see footnote 9 for details). Ina context in which there are 5 guests who arrived, the set of degrees d willbe {1,2,3,4,5}, corresponding to the interval (0,5], and the measure of thatinterval will be 5.

As discussed in §2.1.3, the evaluativity represented by the final require-ment in (31) (that d′ > sq−int) must be contributed by something externalto the quantity word itself. Coupled with the context-sensitivity of the di-mension of measurement built into M-Op, this means that quantity wordconstructions are underspecified with respect to their dimension of measure-ment. Assuming that cardinal, proportional, and reverse proportional quan-tities count as different dimensions of measurement (as do height, length,breadth and depth, under typical assumptions), this context-sensitivity al-lows for just the sort of contextual variation between cardinal, proportional,and reverse proportional interpretations of quantity words reported in §2.1.2(Rett, 2008b).

Rett’s original proposals discuss other details that can’t be addressedhere. For instance, sentences like The many guests arrive involve type-shifted instances of definite determiners in which they range over degrees(Rett 2008b; see also Kotek 2011). And while M-Op is defined in (29)over individuals, the use of quantity words as VP modifiers requires thatit range over eventualities, as well (Nakanishi, 2007; Doetjes, 2007; Rett,2008b, 2014). Finally, although the interval-based definition in (25) allowsquantity words to apply to gradable adjectives in principle, there is reasonto think that constructions like *much tall are prohibited across languagesfor morphological reasons (Doetjes, 1997).

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5.2 Quantity words as predicates of intervals

In a later series of proposals, Solt (2009, 2015) characterizes quantity wordsas gradable predicates of intervals, type 〈d, 〈〈d, t〉, t〉〉.

(32) JmuchK = λdλD[D(d)]

Like Rett’s proposal, Solt’s analyzes quantity words as relations betweena degree and an interval, although she characterizes them as taking theirarguments in the opposite order. This allows her to better highlight theintuitive parallel between quantity words and gradable adjectives, as dis-cussed in §4; like gradable adjectives, quantity words take degrees as theirfirst argument, but the two differ in the nature of their second argument (anindividual for adjectives, an interval for quantity words).

The truth-conditions under both accounts are more or less the same.And they share a number of background assumptions: both employ exis-tential closure over individual arguments in the absence of overt bindingor modification, and both assume that evaluativity is contributed to quan-tity constructions via the same mechanism typically assumed for gradableadjective constructions. Both accounts characterize negative-polar quan-tity words as converses of their positive-antonym counterparts, accountingfor (among other things) split-scope readings observed with few and little(Heim, 2007). Both are explicitly extended to the VP modifier uses of quan-tity words, in which (following work in Nakanishi, 2007; Doetjes, 2007) theyappear to range over events. Both explicitly follow Schwarzschild in assum-ing that the relevant dimension of measurement is contextually valued (andconstrained in terms of monotonicity).

Finally, both employ a null measurement operator to associate entitieswith degree arguments, as in (33), though Solt’s differs slightly from Rett’sM-Op in (29).

(33) JMeasK = λxλd[the salient measure of x is greater than or equal tod]

While Rett assumes an attributive and predicate version of this null mea-surement operator, Solt assumes only the version in (33) and introduces anadditional compositional rule to allow Meas to compose in the prenominalposition.

In Solt’s account, quantity words relate a degree and an interval in termsof set inclusion, rather than in characterizing the former as a measure of thelatter. Solt’s account therefore doesn’t involve the higher-order measure-

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ment that Rett’s does.This is the most significant difference between the two approaches, and

it has real empirical consequences. Rett (2007) argues that higher-ordermeasurement is required for Romanian, for which quantity words are op-tional in quantity questions (as in (34)), with corresponding subtle semanticdifferences.

(34) a. Cıtehow-many-Fpl

femeiwomen

cunoa ste?know-3sg

b. Cıthow-many

deof

multemany-Fpl

femeiwomen

cunoa ste?know-3sg

‘How many women does he know?’

Solt does not address the Romanian data. But higher-order measure-ment does not appear to be at play in English, which allows Solt’s accountmore natural and unified treatment of much-support than Rett’s initial pro-posal did. Much-support, first discussed in Corver (1997), refers to theperplexing fact that while quantity words are prohibited from modifyinglexical adjectives (as in (35-a), Doetjes, 1997), they are required to modifyproadjectives in certain contexts, as in (35-b).

(35) a. Mary is (*much) tall.b. Fred is generous; in fact, he is too *(much) so.

This particular worry about the degree-modifier proposal is explicitly ad-dressed in Rett (2014), where the account is modified to effectively makehigher-order measurement optional.

In an independent vein, Wellwood rejects the claim that adjectives en-code degree arguments, and attributes to quantity words the ability to as-sociate semantic objects with degrees, like M-Op or Meas in previous ac-counts (Wellwood et al., 2012; Wellwood, 2014, 2015). This allows for acompletely parallel and domain-general treatment of quantity words acrossthe constructions covered in §2.2, but it requires nonstandard assumptionsabout the semantics of gradable adjectives, and it is not explicitly extendedto the empirical foci of Rett’s and Solt’s accounts.

6 Summary

In the foundational traditions of first-order logic and Generalized Quanti-fier Theory, quantity words are analyzed as quantifiers, relating two sets ofindividuals. A long-standing competitor to this perspective characterizes

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quantity words as gradable adjectives, i.e. relations between degrees andindividuals. However, as demonstrated in Table 1, both treatments of quan-tity words are unsatisfying for the same reason: they incorrectly predict thatquantity words are only able to range over individuals, when in fact, theyare able to range over events (as VP modifiers); vectors (as PP modifiers);and intervals (as comparative modifiers).

In a series of recent proposals, degree semanticists have argued that all ofthese uses of quantity words have in common that they range over intervals,or sets of degrees. This is transparently true when quantity words modifycomparatives, as in John is much taller than Sue; but, as Rett (2007, 2008b,2014) and Solt (2009, 2015) have argued, the characterization of quantitywords as ranging over intervals also naturally extends to the other usesof quantity words. In addition to these distributional motivations for adomain-general characterization of quantity words, there are a number ofempirical motivations for the switch, including (but not limited to) somephenomena discussed here: the distribution of evaluativity across quantityword constructions (Rett, 2015) and the antonymic relationship betweenpositive quantity words like many and negative ones like few.

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